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2019 Iberoamerican Mathematical Olympiad P5

Source:

September 16, 2019
combinatorics

Problem Statement

Don Miguel places a token in one of the (n+1)2(n+1)^2 vertices determined by an n×nn \times n board. A move consists of moving the token from the vertex on which it is placed to an adjacent vertex which is at most 2\sqrt2 away, as long as it stays on the board. A path is a sequence of moves such that the token was in each one of the (n+1)2(n+1)^2 vertices exactly once. What is the maximum number of diagonal moves (those of length 2\sqrt2) that a path can have in total?
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